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Properties

Label 1782.2.b.a.1781.1
Level $1782$
Weight $2$
Character 1782.1781
Analytic conductor $14.229$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1782,2,Mod(1781,1782)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1782.1781"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1782, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2293416402\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1781.1
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1781
Dual form 1782.2.b.a.1781.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.86370i q^{5} +4.89898i q^{7} -1.00000 q^{8} +3.86370i q^{10} +(-1.73205 - 2.82843i) q^{11} -4.24264i q^{13} -4.89898i q^{14} +1.00000 q^{16} -3.46410 q^{17} +3.34607i q^{19} -3.86370i q^{20} +(1.73205 + 2.82843i) q^{22} +8.62398i q^{23} -9.92820 q^{25} +4.24264i q^{26} +4.89898i q^{28} -1.73205 q^{29} -2.26795 q^{31} -1.00000 q^{32} +3.46410 q^{34} +18.9282 q^{35} +2.92820 q^{37} -3.34607i q^{38} +3.86370i q^{40} -2.53590 q^{41} -0.896575i q^{43} +(-1.73205 - 2.82843i) q^{44} -8.62398i q^{46} +1.27551i q^{47} -17.0000 q^{49} +9.92820 q^{50} -4.24264i q^{52} -5.65685i q^{53} +(-10.9282 + 6.69213i) q^{55} -4.89898i q^{56} +1.73205 q^{58} +9.52056i q^{59} +7.58871i q^{61} +2.26795 q^{62} +1.00000 q^{64} -16.3923 q^{65} -4.92820 q^{67} -3.46410 q^{68} -18.9282 q^{70} -3.20736i q^{71} +4.89898i q^{73} -2.92820 q^{74} +3.34607i q^{76} +(13.8564 - 8.48528i) q^{77} -10.2784i q^{79} -3.86370i q^{80} +2.53590 q^{82} +1.73205 q^{83} +13.3843i q^{85} +0.896575i q^{86} +(1.73205 + 2.82843i) q^{88} +15.9725i q^{89} +20.7846 q^{91} +8.62398i q^{92} -1.27551i q^{94} +12.9282 q^{95} +9.73205 q^{97} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 12 q^{25} - 16 q^{31} - 4 q^{32} + 48 q^{35} - 16 q^{37} - 24 q^{41} - 68 q^{49} + 12 q^{50} - 16 q^{55} + 16 q^{62} + 4 q^{64} - 24 q^{65} + 8 q^{67} - 48 q^{70}+ \cdots + 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) − 3.86370i − 1.72790i −0.503577 − 0.863950i \(-0.667983\pi\)
0.503577 − 0.863950i \(-0.332017\pi\)
\(6\) 0 0
\(7\) 4.89898i 1.85164i 0.377964 + 0.925820i \(0.376624\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.86370i 1.22181i
\(11\) −1.73205 − 2.82843i −0.522233 − 0.852803i
\(12\) 0 0
\(13\) − 4.24264i − 1.17670i −0.808608 − 0.588348i \(-0.799778\pi\)
0.808608 − 0.588348i \(-0.200222\pi\)
\(14\) − 4.89898i − 1.30931i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 3.34607i 0.767640i 0.923408 + 0.383820i \(0.125392\pi\)
−0.923408 + 0.383820i \(0.874608\pi\)
\(20\) − 3.86370i − 0.863950i
\(21\) 0 0
\(22\) 1.73205 + 2.82843i 0.369274 + 0.603023i
\(23\) 8.62398i 1.79822i 0.437718 + 0.899112i \(0.355787\pi\)
−0.437718 + 0.899112i \(0.644213\pi\)
\(24\) 0 0
\(25\) −9.92820 −1.98564
\(26\) 4.24264i 0.832050i
\(27\) 0 0
\(28\) 4.89898i 0.925820i
\(29\) −1.73205 −0.321634 −0.160817 − 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) −2.26795 −0.407336 −0.203668 − 0.979040i \(-0.565286\pi\)
−0.203668 + 0.979040i \(0.565286\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) 18.9282 3.19945
\(36\) 0 0
\(37\) 2.92820 0.481394 0.240697 − 0.970600i \(-0.422624\pi\)
0.240697 + 0.970600i \(0.422624\pi\)
\(38\) − 3.34607i − 0.542803i
\(39\) 0 0
\(40\) 3.86370i 0.610905i
\(41\) −2.53590 −0.396041 −0.198020 − 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) − 0.896575i − 0.136726i −0.997660 − 0.0683632i \(-0.978222\pi\)
0.997660 − 0.0683632i \(-0.0217777\pi\)
\(44\) −1.73205 − 2.82843i −0.261116 − 0.426401i
\(45\) 0 0
\(46\) − 8.62398i − 1.27154i
\(47\) 1.27551i 0.186053i 0.995664 + 0.0930263i \(0.0296541\pi\)
−0.995664 + 0.0930263i \(0.970346\pi\)
\(48\) 0 0
\(49\) −17.0000 −2.42857
\(50\) 9.92820 1.40406
\(51\) 0 0
\(52\) − 4.24264i − 0.588348i
\(53\) − 5.65685i − 0.777029i −0.921443 − 0.388514i \(-0.872988\pi\)
0.921443 − 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) −10.9282 + 6.69213i −1.47356 + 0.902367i
\(56\) − 4.89898i − 0.654654i
\(57\) 0 0
\(58\) 1.73205 0.227429
\(59\) 9.52056i 1.23947i 0.784811 + 0.619736i \(0.212760\pi\)
−0.784811 + 0.619736i \(0.787240\pi\)
\(60\) 0 0
\(61\) 7.58871i 0.971634i 0.874060 + 0.485817i \(0.161478\pi\)
−0.874060 + 0.485817i \(0.838522\pi\)
\(62\) 2.26795 0.288030
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.3923 −2.03322
\(66\) 0 0
\(67\) −4.92820 −0.602076 −0.301038 − 0.953612i \(-0.597333\pi\)
−0.301038 + 0.953612i \(0.597333\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) −18.9282 −2.26235
\(71\) − 3.20736i − 0.380644i −0.981722 − 0.190322i \(-0.939047\pi\)
0.981722 − 0.190322i \(-0.0609532\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) −2.92820 −0.340397
\(75\) 0 0
\(76\) 3.34607i 0.383820i
\(77\) 13.8564 − 8.48528i 1.57908 − 0.966988i
\(78\) 0 0
\(79\) − 10.2784i − 1.15641i −0.815890 − 0.578207i \(-0.803753\pi\)
0.815890 − 0.578207i \(-0.196247\pi\)
\(80\) − 3.86370i − 0.431975i
\(81\) 0 0
\(82\) 2.53590 0.280043
\(83\) 1.73205 0.190117 0.0950586 − 0.995472i \(-0.469696\pi\)
0.0950586 + 0.995472i \(0.469696\pi\)
\(84\) 0 0
\(85\) 13.3843i 1.45173i
\(86\) 0.896575i 0.0966802i
\(87\) 0 0
\(88\) 1.73205 + 2.82843i 0.184637 + 0.301511i
\(89\) 15.9725i 1.69308i 0.532328 + 0.846538i \(0.321317\pi\)
−0.532328 + 0.846538i \(0.678683\pi\)
\(90\) 0 0
\(91\) 20.7846 2.17882
\(92\) 8.62398i 0.899112i
\(93\) 0 0
\(94\) − 1.27551i − 0.131559i
\(95\) 12.9282 1.32641
\(96\) 0 0
\(97\) 9.73205 0.988140 0.494070 − 0.869422i \(-0.335509\pi\)
0.494070 + 0.869422i \(0.335509\pi\)
\(98\) 17.0000 1.71726
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1782.2.b.a.1781.1 ✓ 4
3.2 odd 2 1782.2.b.b.1781.4 yes 4
9.2 odd 6 1782.2.i.j.1187.1 8
9.4 even 3 1782.2.i.l.593.1 8
9.5 odd 6 1782.2.i.j.593.4 8
9.7 even 3 1782.2.i.l.1187.4 8
11.10 odd 2 1782.2.b.b.1781.1 yes 4
33.32 even 2 inner 1782.2.b.a.1781.4 yes 4
99.32 even 6 1782.2.i.l.593.4 8
99.43 odd 6 1782.2.i.j.1187.4 8
99.65 even 6 1782.2.i.l.1187.1 8
99.76 odd 6 1782.2.i.j.593.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.a.1781.1 ✓ 4 1.1 even 1 trivial
1782.2.b.a.1781.4 yes 4 33.32 even 2 inner
1782.2.b.b.1781.1 yes 4 11.10 odd 2
1782.2.b.b.1781.4 yes 4 3.2 odd 2
1782.2.i.j.593.1 8 99.76 odd 6
1782.2.i.j.593.4 8 9.5 odd 6
1782.2.i.j.1187.1 8 9.2 odd 6
1782.2.i.j.1187.4 8 99.43 odd 6
1782.2.i.l.593.1 8 9.4 even 3
1782.2.i.l.593.4 8 99.32 even 6
1782.2.i.l.1187.1 8 99.65 even 6
1782.2.i.l.1187.4 8 9.7 even 3